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Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations. (English) Zbl 0955.65086
Summary: W. R. Madych and S. A. Nelson [Math. Comput. 54, No. 189, 211-230 (1990; Zbl 0859.41004)] proved multiquadric (MQ) mesh-independent radial basis functions (RBFs) enjoy exponential convergence. The primary disadvantage of the MQ scheme is that it is global, hence, the coefficient matrices obtained from this discretization scheme are full. Full matrices tend to become progressively more ill-conditioned as the rank increases.
In this paper, we explore several techniques, each of which improves the conditioning of the coefficient matrix and the solution accuracy. The methods that were investigated are
(1) replacement of global solvers by block partitioning, LU decomposition schemes,
(2) matrix preconditioners,
(3) variable MQ shape parameters based upon the local radius of curvature of the function being solved,
(4) a truncated MQ basis function having a finite, rather than a full band-width,
(5) multizone methods for large simulation problems, and
(6) knot adaptivity that minimizes the total number of knots required in a simulation problem.
The hybrid combination of these methods contribute to very accurate solutions.
Even though the finite element method (FEM) gives rise to sparse coefficient matrices, these matrices in practice can become very ill-conditioned. We recommend using what has been learned from the FEM practitioners and combining their methods with what has been learned in RBF simulations to form a flexible, hybrid approach to solve complex multidimensional problems.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65H10 Numerical computation of solutions to systems of equations
65F35 Numerical computation of matrix norms, conditioning, scaling
35J65 Nonlinear boundary value problems for linear elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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